The Uniqueness Of Meromorphic Functions That Sharing Values

The value distribution theory of meromorphic functions founded by R.Nevanlinna, a famous mathematician, is surely one of the most important achievements in mathematics in the 20th century because not only it is the basis of modern meormorphic function theory, but also it has quite an effect on the development of mathematical branches, and on the interaction among them. Especially, the Nevanlinna theory supplies a very powerful tool to the research of the global analytic solution of complex differential equations. In 1929, R.Nevanlinna studied the conditions with which a meromorphic function can be determined and obtained two celebrated uniqueness theorems for meromorphic functions, which are usully called Nevanlinna's four-value theorem and Nevanlinna's five-value theorem. This launched the investigation of uniqueness theory of meromorphic functions and in particular the shared values of meromorphic functions.The present thesis is part of the author's research work on the shared values of meromorphic functions that concerning derivatives. It consists four chapters.In chapter one, we briefly introduce some main concepts, fundamental results and usual notations concerned with this thesis in the value distribution theory of meromorphic functions.In chapter two, we study the uniqueness problem for entire function sharing one finite nonzero value with its derivatives and obtained the following result, which is a improvement of the theorem given by Hua-Liang Zhong.Theorem 2.1 Let f be a non-constant entire function, n(≥2)be an integer. If f and f 'share a finite nonzero complex number a IM, and when f(z)= a, then f=be^z, where b is a nonzero constant.In chapter three, we study the uniqueness problem for entire function sharing a polynomial with its derivatives and obtained the following results, which improve and extend the theorems given by Xiao-Min Li[12].Theorem 3.1 Let Q j( z )( j = 1,2)be two polynomials and P ( z )be an entire function, k be a positive integer. If f is a non-constant solution of the differential equation thenν( f ) =σ( e^p(z)).Theorem 3.2 Let P ( z )and Q ( z )be two polynomials, n be a positive integer. If f is a transcendental solution of the differential equation such thatν( f)is not a positive integer, and if f ( z )and f ( n)( z )share z IM, then e P ( z)≡1and f =γez,whereγ(≠0)is a finite complex number.In chapter four, we consider the uniqueness problem for meromorphic function f nsharing a small function with f ( k)and obtained the following results, which improve and extend the theorems given by Qing-Cai Zhang[21].Theorem 4.1 Let f be a non-constant meromorphic function, n , k be two positive integers, a ( z )(≡/ 0,∞)be a small function of f . If f n- aand f ( k)- ashare 0 IM and r∈I, whereλis a constant such that 0 <λ< 1,then ff ( kn)--a a≡C for some constant C≠0.Theorem 4.2 Let f be a non-constant meromorphic function, n , k be two positive integers, a ( z )(≡/ 0,∞)be a small function of f . If fn^-a and f^k-a share 0 IM and or fⁿ-a and f^k-a share 0 CM and r∈I, whereλis a constant such that 0 <λ< 1,then f^k≡fⁿ.